A Peano curve is a continuous map $[0,1]\to [0,1]^2$ whose image is the whole square.

I would like to know if on can obtain "holomorphic" Peano curves. Namely, is it possible to find a continuous map $\phi$ from the unit disk $|z|\le 1$ to $\mathbb C^1$ such that
$\phi$ is holomorphic for $|z|<1$ and the image of the boundary $|z|=1$ has non-empty interior in $\mathbb C^1$ under the map $\phi$.

for $|z|<1$, where $\varphi: S^1\to\mathbb{C}$ is a Peano curve (i.e. its image has nonempty interior), and $\phi(z):=\varphi(z)$ for $z\in S^1$. [Edit: this construction doesn't work because $\phi$, as I defined it, may not be continuous up to the boundary - see the comments]

In general there is no reason for this function $\phi$ to be contunuous on $\overline{\mathbb{D}}$! For example, if negative terms are present in the Fourier expansion of $g$, then certainly a continuous extension of the Cauchy Transform of $g$ to the circle will not be equal to $g$.
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Malik YounsiNov 29 '12 at 18:16

2

I see. You are right! I will not delete the answer because it might be an "instructive mistake" (I'm not an analyst, btw) - I will edit adding a disclaimer.
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QfwfqNov 29 '12 at 18:25

@Qfwfq : Yes, it is instructive. Thank you for the edit!
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Malik YounsiNov 30 '12 at 12:48